To evaluate the ability of PFE to capture the spectral energy content of the resulting wavepackets, a comparison with SPOD modes in the wavenumber domain is shown in figure REF . Since results were generated for \(x/D \le 10\) , the spatial Fourier transforms may be contaminated by domain truncation; a Hann window was applied to both SPOD and PFE modes to minimise these effects. For all frequencies, PSE performs reasonably well in capturing the peak wavenumber associated with the wavepackets in this flow, but other energetic wavenumbers observed in the SPOD modes are suppressed due to the PSE regularisation [1]}. The energy of the PSE results at negative wavenumbers (responsible for upstream noise radiation) is several orders of magnitude lower than the peak for low \(St\) , and almost negligible for higher frequencies. While not perfectly aligned, the PFE wavenumber spectra bear several similarities with the SPOD spectra, including the presence of peaks at similar wavenumbers. The absence of upstream waves in the PFE leads to an overall less energetic spectrum for negative wavenumbers, especially for higher frequencies. Still, both PFE and SPOD have a strong energy content at negative propagative wavenumbers (indicated by the dotted lines in figure REF ), suggesting that propagation of PFE wavepackets to the far-field using an acoustic analogy [2]} may lead to the correct directivity [3]} and that BBSAN is generated by wavepacket-shock modulation.
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